Effect of GF errors near threshold

The linear threshold using the 4+2 "thesis closure" is off because of an instability at low k_theta. This weak instability below kinetic threshold (using the thesis closure) connects smoothly to the usual ITG as shown below, for k_theta rho=0.1. Up near R/L_T=6.9 (the base case), the mode's frequency and growth rate agree very well with kinetic theory. Thus this instability near threshold is not a distinct root, but is a low ktheta ITG mode with an inaccurate growth rate. Above the kinetic threshold it becomes an ITG mode with an accurate growth rate, and therefore I don't think that these errors are playing a role in causing our larger chi at the base case parameters.

This "spurious" behavior is only a problem at low k_theta (<0.2). These are very low frequencies, and this mode is sitting very near the branch cut at omega=0, where we cannot match the kinetic response function (it's actually impossible to match exactly since the kinetic response function is discontinuous). Away from marginal (in the upper half omega plane) our linear theory is quite good, so that this problem goes away. The errors we are talking about are pretty small, and thus are only evident near marginal (they're not small compared to zero).


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